Method of Modeling Cloud Point of a Mixture of Fatty Acid Methyl Esters using a Modified UNIFAC Model and a System Therefor

ABSTRACT

A method for predicting onset of liquid phase to solid phase transition of a mixture including a plurality of fatty acid methyl esters components. The method includes identifying chemical and molecular structure of each component of the mixture, calculating activity coefficients for each component in a liquid phase and a solid phase, calculating chemical potential for each component in the liquid phase and in the solid phase at a predetermined temperature and a predetermined pressure, and calculating the cloud point of the mixture. A system for carrying out the method is also disclosed.

The present U.S. Non-provisional patent application is related to and claims the priority benefit of U.S. Provisional Patent Application Ser. No. 61/573,012, filed Aug. 3, 2011, the contents of which is hereby incorporated by reference in its entirety into this disclosure.

TECHNICAL FIELD

The present disclosure generally relates to Fatty Acid Methyl Esters (FAME), and in particular to processing of FAME for use as a constituent of biodiesel.

BACKGROUND

Biodiesel is viewed as the alternative fuel to the petroleum diesel due to the renewable and environmental friendly properties. Biodiesel is a mixture of fatty acid methyl esters (FAME) produced from vegetable oils/animal fats by transesterification with methanol as well as other constituents. The compositions of FAME are constrained by the feedstock of vegetable oils/animal fats. There are six main types of FAME in biodiesel: methyl palmitate (C16:0), methyl palmitoleate (C16:1), methyl stearate (C18:0), methyl oleate (C18:1), methyl linoleate (C18:2) and methyl linolenate (C 18:3); however, there may be other components known to a person having ordinary skill in the art.

The compositions of the FAME significantly affect the cold flow properties. Cold flow properties are the performances of biodiesel at low temperature. Cold flow properties of FAME can be characterized by cloud point, pour point, cold filter plugging point, and low temperature filterability test. Moreover, in North America, cloud point is used as the most appropriate standard to characterize the cold flow properties of FAME. Cloud point is referred as the temperature when biodiesel starts to form crystals (when phase separation begins to appear (i.e., when the mixture becomes “cloudy”) and the thickening fluid can clog filters or other orifices). According to the definition of cloud point, cloud point show FAME change from pure liquid mixture to liquid/solid mixtures. Therefore, cloud point is a phenomenon of solid-liquid equilibrium. The cloud point of FAME depends on the composition because the main FAME components have different melting points (as shown in Table 1). The mixture of FAME with high level of high melting point components will result in a high cloud point.

TABLE 1 Melting point of substantially pure FAME components Components Melting point (° C.) C16:0 30 C16:1 0.5 C18:0 38 C18:1 −20 C18:2 −35 C18:3 −52

The quantitative relationship between the composition of FAME and cloud point is known. For example, Liu et al. have established the quantitative relationship between the composition of FAME and the cloud point through multiple linear statistical regression. This quantitative model shows fatty acid methyl esters with high melting points have more significant effect than those with low melting points. However, the prediction model is challenged due to a low value of R² (proportion of variability in a data set based on how well future outcomes are predicted by a model). Imahara et al. use the thermodynamic phase heterogeneous equilibrium principal to predict the cloud point of fatty acid methyl esters according to the fraction of high melting point component. This prediction model is also challenged because the interaction between the components is not considered. Boros et al. used the thermodynamic model to predict the cloud point of fatty acid methyl esters with the UNIQUAC (UNIversal QUAsiChemical is an activity coefficient model used in description of phase equilibria) to predict the non-ideal behavior and as a result the predictability of the model significantly improved. However, their model is also challenged since it needs to be provided various parameters when a new component is added into a mixture.

While UNIFAC (UNIversal Functional Activity Coefficient) models (see Zhong, Sato, Masuoka, and Chen) have been used for predicting liquid-vapor transitions, the UNIFAC model or the modified UNIFAC model (see Gmehling, Li, and Schiller; Lohmann & Gmehling; Lohmann, Röpke, and Gmehling; Weidlich and Gmehling; and Wittig, Lohmann, and Gmehling) has not been used for predicting liquid-solid transition.

A basic challenge, therefore, remains. Specifically, when various components of fatty acid methyl esters from different sources are added, predicting the cloud point of the new mixture remains a challenge. This challenge is especially problematic since fatty acid methyl esters can originate from many sources. In fact the number of sources from which FAME can originate from may be more diverse than sources of fossil fuel. Furthermore, there can be various additives that can be included in the overall composition. Each of these presents a significant challenge for predicting the cloud point of the mixture.

Therefore, in light of the foregoing challenges with cloud point prediction, a method and a system for accurately predicting cloud point in a mixture of fatty acid methyl esters is needed where the method utilizes molecular interactions between the esters and the relationship therebetween to further provide accuracy to the prediction.

SUMMARY

The present disclosure provides a method for predicting onset of liquid phase to solid phase transition of a mixture including a plurality of fatty acid methyl esters components. The method includes identifying chemical and molecular structure of each component of the mixture. The method further includes calculating activity coefficients for each component in a liquid phase and a solid phase. The method also includes calculating chemical potential for each component in the liquid phase and in the solid phase at a predetermined temperature and a predetermined pressure. The method further includes calculating the cloud point of the mixture

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic of a system for detecting components of a mixture and predicting the cloud point of the mixture.

FIG. 2 is a scheme of enthalpy and entropy change for a mixture transitioning from a liquid to a solid.

FIGS. 3A-3D are plots of cloud point for various binary mixtures of fatty acid methyl esters.

FIG. 4 is a plot of cloud points of a ternary mixture of C14:0/C16:0/C18:0.

FIG. 5 is a plot of cloud points of a ternary mixture of C16:0/C18:0/C18:1.

FIG. 6 is a plot of cloud points of a ternary mixture of C18:1/C18:2/C18:3.

FIG. 7 is a plot of predicted cloud points vs. detected cloud points for the method according to the present disclosure and the methods presented in the prior art.

FIG. 8 is a flow chart of a method according to the present disclosure.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of the present disclosure, reference will now be made to the embodiments illustrated in the drawings, and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of this disclosure is thereby intended.

A novel method and system have been developed for predicting cloud point in a mixture including fatty acid methyl esters (FAME). The system 100 is depicted in FIG. 1. In the system 100, a component detection unit 110; e.g. a mass spectrometer (i.e., a unit which uses masses of particles for determining the elemental composition of a sample) is coupled to a cloud point modeling unit 120. The component detection unit 110 is configured to receive samples (identified as COMP_(i) in FIG. 1, referring to multiple samples from components of a mixture, e.g., a biodiesel mixture) sequentially. The samples are analyzed by the component detection unit 110 and molecular information of each component is provided to the cloud point modeling unit 120. The cloud point modeling unit 120 is configured to receive molecular information (identified as FAME_(i) in FIG. 1, indicating sequential transfer of molecular information associated with each component COMP_(i)) from the component detection unit 110 and further configured to use the method outlined below to predict the cloud point of a mixture that includes the components. It should be appreciated that parts of the system depicted in FIG. 1 can be operated sequentially or in a parallel fashion. For example, a sample of the mixture intended to form the biodiesel mixture can be provided to the component detection unit 110 as a single sample, and the component detection unit 110 can provide molecular information about individual components of the mixture to the cloud point modeling unit 120 in a sequential manner.

Prediction of Cloud Point

To describe the cloud point modeling methodology, a theoretical description of molecular interaction is provided herein.

Phase Equilibrium in the Heterogeneous Closed System

For a closed system with n phases and m components, at equilibrium, there exist the following relations:

(1) The temperature in each phase is the same:

T ₁ =T ₂ = . . . =T _(n) =T  Eq. 1

(2) The pressure in each phase is the same:

P ₁ =P ₂ = . . . =P _(n) =P  Eq. 2

(3) The chemical potential (i.e., partial molar free energy, is a form of potential energy that can be absorbed or released during a chemical reaction) of component i in each phase (i.e., liquid and solid) is the same:

μ_(i) ¹=μ_(i) ²= . . . =μ_(i) ^(n)=μ_(i)  Eq. 3

The chemical potential of each component i is represented by:

μ_(i)(T,P)=μ_(i) ^(o)(T,P)+RT ln(α_(i))  Eq. 4

Where

μ_(i) ^(o): Standard chemical potential at temperature T and pressure P R: gas constant with the value of 8.3145 J·mol⁻¹·K⁻¹ T: is the temperature α_(i): Activity of component i The chemical potential of A in phase 1 and phase 2 are shown in 5 and 6, respectively.

μ_(A) ¹(T,P)=μ_(A) ^(1,o)(T,P)+RT ln(α_(A) ¹)  Eq. 5

Where

μ_(A) ¹(T,P): Chemical potential of A in phase 1 at temperature T and pressure P μ_(A) ^(1,o)(T,P): Standard chemical potential of A in phase 1 at temperature T and pressure P α_(A) ¹: Activity of A in phase 1

μ_(A) ²(T,P)=μ_(A) ^(2,o)(T,P)+RT ln(α_(A) ²)  Eq. 6

Where

μ_(A) ²(T,P): Chemical potential of A in phase 2 at temperature T and pressure P μ_(A) ^(2,o)(T,P): Standard chemical potential of A in phase 2 at temperature T and pressure P μ_(A) ²: Activity of A in phase 2

When A in phase 1 and phase 2 are in an equilibrium state, the chemical potential of A in each phase is the same. Therefore, Eq. 5 and Eq. 6 can be combined into Eq. 3, as provided below in Eq. 7.

μ_(A) ^(1,o)(T,P)RT ln(α_(A) ¹)=μ_(A) ^(2,o)(T,P)RT ln(α_(A) ²)  Eq. 7

The relationship between the activity of A in phase 1 and phase 2 is shown in Eqs. 8 and 9.

RT ln(α_(A) ¹/α_(A) ²)=Δμ_(A) ^(o)(T,P)  Eq. 8

Δμ_(A) ^(o)(T,P)=μ_(A) ^(2,o)(T,P)−μ_(A) ^(1,o)(T,P)  Eq. 9

Where

Δμ_(a) ^(O)(T,p): Standard chemical potential change of A from phase 1 to 2

Therefore, according to condition of heterogeneous phase equilibrium and the definition of chemical potential, Eqs. 8 and 9 can be written in the form of Eq. 9A.

$\begin{matrix} {{{RT}\; {\ln \left( \frac{\gamma_{i}^{S}x_{i}^{S}}{\gamma_{i}^{L}x_{i}^{L}} \right)}} = {{\mu_{i}^{0,L}\left( {T,P} \right)} - {\mu_{i}^{0,S}\left( {T,P} \right)}}} & {{{Eq}.\mspace{14mu} 9}A} \end{matrix}$

The chemical potential cannot be readily calculated; however, it can be calculated according the following relationship (as shown in Eq. 10).

Δμ_(A) ^(o)(T,P)=ΔH _(m) −TΔS _(m)  Eq. 10

Where

ΔH_(m): Enthalpy change of A from phase 1 to 2 ΔS_(m): Entropy change of A from phase 1 to 2

The enthalpy and entropy changes are state variables in thermodynamics and can be calculated by designing a calculable route. An example of such a route is provided in FIG. 2. Therefore, at the heterogeneous phase equilibrium, there is the following relationship.

RT ln(α_(A) ¹/α_(A) ²)=ΔH _(m) −TΔS _(m)  Eq. 11

To calculate the enthalpy change and entropy change for component i from liquid to solid, a new route is designed and consists of three steps. The liquid component i at temperature T and pressure P is chilled to the melting point of component i (T_(m,i)) and the enthalpy change and entropy change are ΔH_(m,i) ¹(T,P) and ΔS_(m,i) ¹(T,P). The liquid component i changes from liquid to solid at the melting point of component i, and the enthalpy change and entropy change are ΔH_(m,i) ²(T,P) and ΔS_(m,i) ²(T,P). Solid component i is heated from the melting point of component i to temperature T and the enthalpy change and entropy change are ΔH_(m,i) ¹(T,P) and ΔS_(m,i) ¹(T,P). The enthalpy change and entropy change in each step are shown in Eq. 11A to 11F.

ΔH _(m,i) ¹(T,P)=∫_(T) ^(T) ^(m,i) C _(p,m,i) ^(L) dT  Eq. 11A

ΔS _(m,i) ¹(T,P)=∫_(T) ^(T) ^(m,i) C _(p,m,i) ^(L) dT  Eq. 11B

ΔH _(m,i) ²(T,P)=−Δ_(fus) H _(m,i)  Eq. 11C

ΔS _(m,i) ²(T,P)=−Δ_(fus) H _(m,i) /T _(m,i)  Eq. 11D

ΔH _(m,i) ³(T,P)=∫_(T) _(m,i) ^(T) C _(p,m,i) ^(S) dT  Eq. 11E

ΔS _(m,i) ³(T,P)=∫_(T) _(m,i) ^(T) C _(p,m,i) ^(S) /TdT  Eq. 11F

Where

C_(p,m,i) ^(L): Molar heat capacity of liquid component i at constant pressure C_(p,m,i) ^(S): Molar heat capacity of solid component i at constant pressure Δ_(fus)H_(m,i): Molar fusion enthalpy of component i

According to thermodynamic state variables, there are the following two relations as shown in Eqs. 11G and 11H.

ΔH _(m,i) ^(o)(T,P)=ΔH _(m,i) ¹(T,P)+ΔH _(m,i) ²(T,P)+ΔH _(m,i) ³(T,P)  Eq. 11G

ΔS _(m,i) ^(o)(T,P)=ΔS _(m,i) ¹(T,P)+ΔS _(m,i) ²(T,P)+ΔS _(m,i) ³(T,P)  Eq. 11H

According to Eqs. 11A through 11H, the enthalpy change and entropy change of component i from liquid to solid are shown in Eq. 11I and 11J.

ΔH _(m,i) ^(o)(T,P)=−∫_(T) ^(T) ^(m,i) ΔC _(p,m,i) dT−Δ _(fus) H _(m,i)  Eq. 11I

ΔS _(m,i) ^(o)(T,P)=−∫_(T) ^(T) ^(m,i) (ΔC _(p,m,i) /T)dT−Δ _(fus) H _(m,i) /T _(m,i)  Eq. 11J

Where

ΔC_(p,m,i): Molar heat capacity difference of component i at constant pressure in liquid and solid

ΔC _(p,m,i) =C _(p,m,i) ^(L) −C _(p,m,i) ^(S)  Eq. 11K

According to the above equations, one thermodynamic model to predict the cloud point as the function of the composition is show in Eq. 11L.

RT ln(α_(i) ^(L)/α_(i) ^(S))=Δ_(fus) H _(m,i)(1−T/T _(m,i))−∫_(T) ^(T) ^(m,i) ΔC _(p,m,i) dT+T∫ _(T) ^(T) ^(m,i) (ΔC _(p,m,i) /T)dT  Eq. 11L

The heat capacity difference of component i in liquid and solid can be considered negligible and the thermodynamic model can then be expressed according to Eq. 11M.

RT ln(Δ_(i) ^(L)/Δ_(i) ^(S))=−Δ_(fus) H _(m,i)(1−T/T _(m,i))  Eq. 11M

The solid only contains one component in an ideal solution. Therefore, the thermodynamic model can be expressed according to Eq. 11N.

RT ln Δ_(i) ^(L)=−Δ_(fus) H _(m,i)(1−T/T _(m,i))  Eq. 11N

According to the definition of activity (further defined herein), the thermodynamic model changes to Eq. 11O.

R ln(γ_(i) ^(L)χ_(i) ^(L))=Δ_(fus) H _(m,i)(1/T−1/T)  Eq. 11O

The activity coefficient of the component in the mixture of FAME can be calculated according to the Modified Universal Functional Activity Coefficient (UNIFAC) model, further described below. For a given composition of FAME, there is a calculated temperature according to Eq. 11O for each component. The cloud point of the mixture of FAME is the highest calculated temperature.

In a special case, the mixture of FAME is close to an ideal solution. The activity coefficient is 1 and the thermodynamic model becomes to Eq. 11P.

R ln(χ_(i) ^(L))=Δ_(fus) H _(m,i)(1/T _(m,i)−1/T)  Eq. 11P

For a given composition of FAME, a temperature is calculated according to Eq. 11P for each component. The highest calculated temperature is the cloud point of the mixture of FAME.

Modified Universal Functional Activity Coefficient (UNIFAC) Model

As seen in Eq. 11, activities are introduced to the model. The activity of A is defined as in Eq. 12.

α_(A)=γ_(A)χ_(A)  Eq. 12

Where

γ_(A): Activity coefficient of A χ_(A): Mole fraction of A

When the components are independent and do not interact, the system is ideal. Therefore, the activity coefficient is 1 and the activity is equal to the molar fraction. Thus, for the ideal system, the thermodynamic model for the heterogeneous phase equilibrium is shown in Eq. 13.

RT ln(χ_(A) ¹/χ_(A) ²)=ΔH _(m) −TΔS _(m)  Eq. 13

According to the relationship between chemical potential and enthalpy/entropy, the thermodynamic model is written as

$\begin{matrix} {{{{RT}\; {\ln \left( \frac{\gamma_{i}^{S}x_{i}^{S}}{\gamma_{i}^{L}x_{i}^{L}} \right)}} = {{\Delta \; {H_{m,i}\left( {T,P} \right)}} - {T\; \Delta \; {S_{m,i}\left( {T,P} \right)}}}}{with}} & {{{Eq}.\mspace{14mu} 13}A} \\ {{\Delta \; H_{m,i}{\int_{T}^{T_{m,i}}{\Delta \; C_{P,m,i}\ {T}}}} + {\Delta_{fus}H_{m,i}}} & {{{Eq}.\mspace{14mu} 13}B} \\ {{\Delta \; S_{m,i}} = {\frac{\Delta_{fus}H_{m,i}}{T_{m,i}} + {\int_{T}^{T_{m,i}}{\left( \frac{\Delta \; C_{P,m,i}}{T} \right)\ {T}}}}} & {{{Eq}.\mspace{14mu} 13}C} \end{matrix}$

Where

Δ_(fus)H_(m,i) and ΔC_(p,m,i) are the molar fusion enthalpy of component i and the difference in the heat capacity at constant pressure between solid phase and liquid phase, respectively, and T_(m,i) is the melting point of component i.

Therefore, the thermodynamic model can be provided as

$\begin{matrix} {{{RT}\; {\ln \left( \frac{\gamma_{i}^{S}x_{i}^{S}}{\gamma_{i}^{L}x_{i}^{L}} \right)}} = {{\int_{T}^{T_{m,i}}{\Delta \; C_{P,m,i}\ {T}}} + {\Delta_{fus}{H_{m,i}\left( {1 - \frac{T}{T_{m,i}}} \right)}} - {T{\int_{T}^{T_{m,i}^{\prime}}{\left( \frac{\Delta \; C_{P,m,i}}{T} \right)\ {T}}}}}} & {{{Eq}.\mspace{14mu} 13}D} \end{matrix}$

The heat capacity at constant pressure change from solid phase to liquid phase is small enough to be neglected. Thus, the thermodynamic model can be provided as

$\begin{matrix} {{\ln \left( \frac{\gamma_{i}^{S}x_{i}^{S}}{\gamma_{i}^{L}x_{i}^{L}} \right)} = {\frac{\Delta_{fus}H_{m,i}}{R}\left( {\frac{1}{T} - \frac{1}{T_{m,i}}} \right)}} & {{{Eq}.\mspace{14mu} 13}E} \end{matrix}$

Generally, the solid phase has small amount of fatty acid methyl esters at the cloud point. Therefore, the solid phase can be viewed as one component and an ideal solution. Consequently, the thermodynamic model is written as

$\begin{matrix} {{\ln \left( {\gamma_{i}^{L}x_{i}^{L}} \right)} = {\frac{\Delta_{fus}H_{m,i}}{R}\left( {\frac{1}{T_{m,i}} - \frac{1}{T}} \right)}} & {{{Eq}.\mspace{14mu} 13}F} \end{matrix}$

This equation is used to calculate the T for different components and the maximum value. of T is viewed as the cloud point.

When the mixture of fatty acid methyl esters is viewed as an ideal solution, the activity coefficient is 1 and the model is written as

$\begin{matrix} {{\ln \left( x_{i}^{L} \right)} = {\frac{\Delta_{fus}H_{m,i}}{R}{\left( {\frac{1}{T_{m,i}} - \frac{1}{T}} \right).}}} & {{{Eq}.\mspace{14mu} 13}G} \end{matrix}$

However, modeling using ideal framework results in unacceptable inaccuracies. Therefore, it is necessary to know the activity coefficient in non-ideal systems for the utilization of the thermodynamic model of heterogeneous phase equilibrium. The modified UNIFAC model is the most accurate for calculating the activity coefficients. The modified UNIFAC model is derived from UNIFAC model.

To further describe the modified UNIFAC model, first the UNIFAC model is described. In the UNIFAC model, the activity coefficient has two parts: the effect of the group shape and the effect of the group interactions (as shown in Eq. 14).

ln γ_(i)=ln γ_(i) ^(GS)+ln γ_(i) ^(GI)  Eq. 14

Where

γ_(i): Activity coefficient of component i γ_(i) ^(GS): Effect of group interaction on the activity coefficient of component i γ_(i) ^(GI): Effect of group interaction on the activity coefficient of component i The effect of the group shape on the activity coefficient is expressed in Eq. 15.

ln γ_(i) ^(GS)=1−V ₁+ln V _(i)−5q _(i)(1−V _(i) /F _(i)+ln(V _(i) /F _(i)))  Eq. 15

Where

V _(i) =r _(i) /Σ _(j)χ_(j) r _(j)  Eq. 16

r _(i)=Σ_(i)ν_(ki)δ_(i)  Eq. 17

F _(i) =q _(i)/Σ_(j)χ_(j) q _(j)  Eq. 18

q _(i)=Σ_(i)ν_(ki) Q _(i)  Eq. 19

Where

χ_(j): Mole Fraction of component j δ_(k): Volume parameter of group k Q_(k): Surface area parameter of group k ν_(ki): Number of group k in component i

The effect of the group interaction on the activity coefficient is shown in Eq. 20.

ln γ_(i) ^(GI)=Σ_(k)ν_(ki)(ln η_(k)−ln η_(k) ^(i))  Eq. 20

ln η_(k) is the group k contribution on the activity coefficient through the group interaction (as shown in Eq. 21) and ln η_(k) ^(i) is the group k contribution on the activity coefficient through the group interaction in the pure component i (as shown in Eq. 22).

ln η_(k)=5Q _(k)(1−ln(Σ_(m)θ_(m)τ_(mk))−Σ_(i)(θ_(i)τ_(ki))/Σ_(j)θ_(j)τ_(ji))  Eq. 21

ln η_(k) ^(i)=5Q _(k)(1−ln(Σ_(m)θ_(m)τ_(mk))−Σ_(i)(θ_(i)τ_(ki))/Σ_(j)θ_(j)τ_(ji))(for x _(i)=1)  Eq. 22

Where

θ_(m) =Q _(m) X _(m)/Σ_(n) Q _(n) X _(n)  Eq. 23

X _(m)=Σ_(j)ν_(mj)χ_(j)/Σ_(n)Σ_(j)ν_(mj)χ_(j)  Eq. 24

τ_(m)=exp(−A _(ji) /T)  Eq. 25

Where

A_(ji): Group interaction parameter

To decrease the deviation in predicting activity coefficient, the UNIFAC model was modified. According to the modified UNIFAC model, the activity coefficient includes two parts: the effect of the group shape on the activity coefficient and the effect of the group interaction on the activity coefficient. In the modified UNIFAC model, both the effects of group shape and group interaction on the activity coefficient of the modified UNIFAC model are different from those of the UNIFAC model.

According to the modified UNIFAC model, the effect of the group shape on the activity coefficient is expressed in Eq. 26.

ln γ_(i) ^(GS)=1−V _(i) ¹+ln V _(i) ¹−5q _(i)(1−V _(i) /F _(i)+ln(V _(i) /F _(i)))  Eq. 26

And

V _(i) ¹ =r _(i) ^(3/4)/Σ_(j)χ_(j) r _(j) ^(3/4)  Eq. 27

V _(i) =r _(i)/Σ_(j)χ_(j) r _(j)  Eq. 28

r _(i)=Σ_(i)ν_(ki)δ_(i)  Eq. 29

F _(i) =q _(i)/Σ_(j)χ_(j) q _(j)  Eq. 30

q _(i)=Σ_(i)ν_(ki) Q _(i)  Eq. 31

Where

χ_(j): Molar Fraction of component j δ_(k): Volume parameter of group k Q_(k): Surface area parameter of group k ν_(ki): Number of group k in component i

The effect of the group interaction on the activity coefficient is shown in Eq. 32.

ln γ_(i) ^(GI)Σ_(k)ν_(ki)(ln η_(k)−ln η_(k) ^(i))  Eq. 32

where, ln η_(k) is the group k contribution on the activity coefficient through the group interaction (as shown in Eq. 33), ln η_(k) ^(i) is the group k contribution on the activity coefficient through the group interaction in the pure component i (as shown in Eq. 34).

$\begin{matrix} {{\ln \; \eta_{k}} = {\frac{z\; Q_{k}}{2}\left\{ {{- {\ln\left( {\sum\limits_{m}{\theta_{m}\tau_{mk}}} \right)}} + 1 - \frac{\sum\limits_{l}{\theta_{l}\tau_{kl}}}{\sum\limits_{j}{\theta_{j}\tau_{jl}}}} \right\}}} & {{Eq}.\mspace{14mu} 33} \\ {{{\ln \; \eta_{k}^{i}} = {\frac{z\; Q_{k}}{2}\left\{ {{- {\ln\left( {\sum\limits_{m}{\theta_{m}\tau_{mk}}} \right)}} + 1 - \frac{\sum\limits_{l}{\theta_{l}\tau_{kl}}}{\sum\limits_{j}{\theta_{j}\tau_{jl}}}} \right\}}}\left( {{{for}\mspace{14mu} x_{i}} = 1} \right){where}} & {{Eq}.\mspace{14mu} 34} \\ {\theta_{m} = \frac{Q_{m}X_{m}}{\sum\limits_{n}{Q_{n}X_{n}}}} & {{Eq}.\mspace{14mu} 35} \\ {X_{m} = \frac{\sum\limits_{j}{\upsilon_{mj}x_{j}}}{\sum\limits_{n}{\sum\limits_{j}{\upsilon_{nj}x_{j}}}}} & {{Eq}.\mspace{14mu} 36} \\ {\tau_{ji} = {\exp \left( {{- \frac{A_{ji}}{T}} - B_{ji} - {C_{ji}T}} \right)}} & {{Eq}.\mspace{14mu} 37} \end{matrix}$

where A_(ji): Group interaction parameter B_(ji): Group interaction parameter C_(ji): Group interaction parameter

To apply the thermodynamic model disclosed herein to predict the cloud point according to the composition of fatty acid methyl esters, the properties of pure fatty acid methyl esters such as melting points and fusion enthalpy should be known. To use the modified UNIFAC model in activity coefficients prediction, the group shape parameters and group interaction parameter should be known. These parameters are discussed below.

Parameters Melting Points and Fusion Enthalpy

To predict the cloud point based on the composition of fatty acid methyl esters by the above thermodynamic model, the melting points and fusion enthalpies of the pure components should be known. The relationship between melting point and fusion enthalpy as shown in Eq. 38.

Δ_(fus) H _(m) =T _(m)Δ_(fus) S _(m)  Eq. 38

The fusion enthalpy and fusion entropy can be calculated by a group contribution model according to Eq. 39.

$\begin{matrix} {{\Delta_{fus}S_{m}} = {\sum\limits_{i}{n_{i}\kappa_{i}}}} & {{Eq}.\mspace{14mu} 39} \end{matrix}$

Where n_(i) is the number of group i in the component, and is the group value of entropy contribution, respectively.

According to the group contribution model, the fatty acid methyl esters have the following groups: —CH₃, —CH₂—CH═ and —C(═O)O—. The group contributions for fusion enthalpy are shown in Table 2.

TABLE 2 Group values for the fusion entropy contributions Group CH₃— —CH₂— —CH═ —(C=O)O— Group Values 17.6 7.1^(a) 5.3 7.7 J · mol⁻¹ · K⁻¹ ^(a)The group value will multiply 1.31 for the number of consecutive methylene groups no less than the sum of the remaining groups.

The melting points of fatty acid methyl esters are shown in Table 1. The fusion enthalpies of fatty acid methyl esters are shown in Table 3.

TABLE 3 Fusion enthalpies of fatty acid methyl esters Components C16:0 C16:1 C18:0 C18:1 C18:2 C18:3 Δ_(fus)H_(m, i) 55350 — 64430 43890 — — (experimental) Δ_(fus)H_(m, i) 52480 43733 59844 46507 41846 37089 (predicted)

The fusion enthalpies of the saturated pure fatty acid methyl esters were determined. Due to the non-ideal property of the mixture of fatty acid methyl esters, the activity coefficients of the components are determined. For the methyl esters, according the modified UNIFAC model, the groups include CH₂, CH₃, CH═CH and (C═O)OCH₃. The group shape parameters are shown in Table 4 and the group interaction parameters are shown in Table 5. Based on the composition of fatty acid methyl esters and the group parameter, above equations can be used to predict the activity coefficients.

TABLE 4 Group Shape parameters in the modified UNIFAC model CH₃ CH₂ CH═CH (C═O)OCH₃ δ_(k) 0.6325 0.6325 1.2832 1.2700 Q_(k) 1.0608 0.7081 1.2489 1.6286

TABLE 5 Group interaction parameters in the modified UNIFAC model CH2/CH3 CH═CH (C═O)OCH₃ CH2/CH3 A 0 189.66 98.656 B 0 −0.2723 1.9294 C 0 0 3.133 × 10⁻³ CH═CH A −95.418 0 980.74 B 6.171 × 10⁻² 0 −2.4224 C 0 0 0 (C═O)OCH₃ A 632.22 −582.82 0 B −3.3912 1.6732 0 C 3.928 × 10⁻³ 0 0

Results Cloud Points of a Binary System

The model was tested for several binary FAME and the measured and predicted cloud points of these binary mixtures are shown in FIG. 3. For the binary mixture of saturated FAME components, there are eutectic points (see FIGS. 3A, 3C and 3D). For example, in the case of the mixture of C12:0/C14:0, the cloud point of the mixture decreased to the eutectic point (approximately. 70% of C 12:0). Binary mixtures of C12:0/C16:0, C12:0/C18:0, C14:0/C16:0, C14:0/C18:0/and C16:0/C18:0 showed similar behavior (see FIGS. 3A, 3C and 3D). Binary mixtures of saturated/unsaturated FAME components, such as the mixtures of C18:1/C16:0 and C18:1/C18:0 did not show eutectic compositions (see FIG. 3B). The cloud points of these binary systems increased sharply with the fraction of the saturated FAME when the fraction of C18:1 is larger than 80%. For the binary mixture of C18:1/C18:2, there is also eutectic point.

Cloud Points and the Compositions in Ternary System

Ternary systems were examined composed of: C14:0/C16:0/C18:0, C18:/C18:2/C18:3 and C16:0/C18:0/C18:1. The predicted and experimentally measured cloud points of these ternary mixtures are presented in FIGS. 4, 5, and 6. The surfaces of these figures consist of the predicted cloud points and the balls in these figures are the measured cloud points. The ternary mixtures of C14:0/C16:0/C18:0 and C18;1/C18:2/C18:3 have a ternary eutectic point (FIGS. 4 and 6). The cloud point of the mixture at the ternary eutectic point is lowest. The ternary mixture of C16:0/C18:0/C18:1 does not have a ternary eutectic point (FIG. 5).

Using method discussed in the present disclosure, the predicted and experimentally measured cloud points are plotted in FIG. 7. The correlation of the predicted and measured cloud points depicts a good correlation with a linear relationship (Eq. 40) with R² as high as 0.99 between predicted and detected cloud point.

T _(CP,P)=0.975T _(CP,D)+8.55  Eq. 40

Where T_(CP,P) and T_(CP,D) are the predicted cloud points and detected cloud points of the mixtures of FAME, respectively.

In operation, referring back to FIG. 1, the cloud point modeling unit 120 can include an application specific integrated circuit or a computer. In each case, a memory (not shown) can be used to hold both i) executable software code prepared from a source code and ii) scratchpad memory for necessary calculations. A software implementation using NetBeans IDE compiler and a package from e j technologies for compiling and packaging the software for modeling cloud point based on the method disclosed herein is provided in Appendix-A, filed herewith, entirety of which is incorporated herein by reference.

The method described herein is depicted in FIG. 8. The method 200 is for predicting onset of liquid phase to solid phase transition of a mixture including a plurality of fatty acid methyl esters components. The method 200 includes:

identifying chemical and molecular structure of each component of the mixture (step 210). The method 200 also includes calculating activity coefficients for each component in a liquid phase and a solid phase according to

${\ln \; \gamma_{i}^{GS}} = {1 - V_{i}^{\prime} + {\ln \; V_{i}^{\prime}} - {5{q_{i}\left( {1 - \frac{V_{i}}{F_{i}} + {\ln \left( \frac{V_{i}}{F_{i}} \right)}} \right)}}}$ and ${\ln \; \gamma_{i}^{GI}} = {\sum\limits_{k}{\upsilon_{ki}\left( {{\ln \; \eta_{k}} - {\ln \; \eta_{k}^{i}}} \right)}}$

(step 220). The method 200 also includes: calculating chemical potential for each component in the liquid phase and in the solid phase at a predetermined temperature and a predetermined pressure according to

μ_(i) ^(L)=μ_(i) ^(0,L)(T,P)+RT ln(γ_(i) ^(L)χ_(i) ^(L))

μ_(i) ^(S)=μ_(i) ^(0,S)(T,P)+RT ln(γ_(i) ^(S)χ_(i) ^(S))

(step 230). The method 200 also includes calculating the cloud point of the mixture (step 240).

Those skilled in the art will recognize that numerous modifications can be made to the specific implementations described above. Therefore, the following claims are not to be limited to the specific embodiments illustrated and described above. The claims, as originally presented and as they may be amended, encompass variations, alternatives, modifications, improvements, equivalents, and substantial equivalents of the embodiments and teachings disclosed herein, including those that are presently unforeseen or unappreciated, and that, for example, may arise from applicants/patentees and others. 

1. A method for predicting onset of liquid phase to solid phase transition of a mixture including a plurality of fatty acid methyl esters components, comprising: identifying chemical and molecular structure of each component of the mixture; calculating activity coefficients for each component in a liquid phase and a solid phase according to ln ${\ln \; \gamma_{i}^{GS}} = {1 - V_{i}^{\prime} + {\ln \; V_{i}^{\prime}} - {5{q_{i}\left( {1 - \frac{V_{i}}{F_{i}} + {\ln \left( \frac{V_{i}}{F_{i}} \right)}} \right)}}}$ and ${{\ln \; \gamma_{i}^{GI}} = {\sum\limits_{k}{\upsilon_{ki}\left( {{\ln \; \eta_{k}} - {\ln \; \eta_{k}^{i}}} \right)}}};$ calculating chemical potential for each component in the liquid phase and in the solid phase at a predetermined temperature and a predetermined pressure according to μ_(i) ^(L)=μ_(i) ^(0,L)(T,P)+RT ln(γ_(i) ^(L)χ_(i) ^(L)) and μ_(i) ^(S)=μ_(i) ^(0,S)(T,P)+RT ln(γ_(i) ^(S)χ_(i) ^(S)); and calculating the cloud point of the mixture according to RT ln ${\left( \frac{\gamma_{i}^{S}x_{i}^{S}}{\gamma_{i}^{L}x_{i}^{L}} \right) = {{\mu_{i}^{0,L}\left( {T,P} \right)} - {\mu_{i}^{0,S}\left( {T,P} \right)}}},$ wherein ${V_{i}^{\prime} = \frac{r_{i}^{3/4}}{\sum\limits_{j}{x_{j}r_{j}^{3/4}}}},{V_{i} = \frac{r_{i}}{\sum\limits_{j}{x_{j}r_{j}}}},{r_{i} = {\sum{v_{ki}\delta_{k}}}},{F_{i} = \frac{q_{i}}{\sum{x_{i}q_{i}}}},{q_{i} = {\sum{v_{ki}Q_{k}}}},{{\ln \; \eta_{k}} = {\frac{z\; Q_{k}}{2}\left\{ {{- {\ln\left( {\sum\limits_{m}{\theta_{m}\tau_{mk}}} \right)}} + 1 - \frac{\sum\limits_{l}{\theta_{l}\tau_{kl}}}{\sum\limits_{j}{\theta_{j}\tau_{jl}}}} \right\}}},{{\ln \; \eta_{k}^{i}} = {\frac{z\; Q_{k}}{2}{\left\{ {{- {\ln\left( {\sum\limits_{m}{\theta_{m}\tau_{mk}}} \right)}} + 1 - \frac{\sum\limits_{l}{\theta_{l}\tau_{kl}}}{\sum\limits_{j}{\theta_{j}\tau_{jl}}}} \right\}'}}}$ ${\theta_{m} = \frac{Q_{m}X_{m}}{\sum\limits_{n}{Q_{n}X_{n}}}},{X_{m} = \frac{\sum\limits_{j}{\upsilon_{mj}x_{j}}}{\sum\limits_{n}{\sum\limits_{j}{\upsilon_{nj}x_{j}}}}},{\tau_{ji} = {\exp \left( {{- \frac{A_{ji}}{T}} - B_{ji} - {C_{ji}T}} \right)}},$ and wherein group shape parameters δ_(k) and Q_(k) and group interaction parameters A, B, and C are determined for each type of bond for each component of the mixture. 